Select the new equation formed after collecting all variables on one side of the equal sign. (There is more than 1 correct answer.)

1/2(6x + 8) = x − 8

Responses

4 = -2x - 8
4 = -2x - 8

3x = x -12
3x = x -12

8 = -2x - 8
8 = -2x - 8

2x + 4 = -8
2x + 4 = -8

3x + 12 = x
3x + 12 = x

2x + 8 = 8

1 answer

To collect all variables on one side of the equation \( \frac{1}{2}(6x + 8) = x - 8 \), let's first simplify the left-hand side:

  1. Distribute \( \frac{1}{2} \): \[ \frac{1}{2}(6x + 8) = 3x + 4 \]

Now we have the equation: \[ 3x + 4 = x - 8 \]

Next, we will rearrange it by moving all terms involving \( x \) to one side and all constant terms to the other side. We can do this in different ways:

  1. Subtract \( x \) from both sides: \[ 3x - x + 4 = -8 \] This simplifies to: \[ 2x + 4 = -8 \] (Correct Answer)

  2. Subtract \( 4 \) from both sides: \[ 3x + 4 - 4 = x - 8 - 4 \] This simplifies to: \[ 3x = x - 12 \] (Correct Answer)

  3. Subtract \( 4 \) from both sides and then increase both sides by \( 8 \): \[ 3x + 4 - 4 + 8 = x - 8 + 8 \] This simplifies to: \[ 3x + 12 = x \] (Correct Answer)

  4. Rearranging the equation can also lead us to: \[ 3x + 4 + 8 = x \] This simplifies to: \[ 8 = -2x - 8 \] (Correct Answer, if we apply a different approach)

From your responses, the correct answers after rearranging would be:

  • \( 2x + 4 = -8 \)
  • \( 3x = x - 12 \)
  • \( 3x + 12 = x \)

So the correct answers based on these steps are:

  1. \( 2x + 4 = -8 \)
  2. \( 3x = x - 12 \)
  3. \( 3x + 12 = x \)

Note: The other responses provided (like \( 4 = -2x - 8 \) and \( 8 = -2x - 8 \)) do not appear to be in a consistent arrangement based on the starting equation.